Asymmetric high speed Digital Subscriber Line (ADSL) and Very high speed Digital Subscriber Line (VDSL) are examples of modem communication systems which permit transmission of data over band limited communication lines at very high rates, e.g., up to 52 Mbits/s—see for example “ADSL, VDSL and Multicarrier Modulation”, J. A. C. Bingham, Wiley, 2000. They are examples of multi-carrier systems. Multi-carrier modulation is well known for transmitting digital data by splitting the data into fixed-length data blocks or symbols each having the same number of sub-blocks or bits. Analog transmission of these blocks is carried out using a set of carrier signals. There is a carrier for each of the sub-blocks in one block. The carriers have frequencies that are equally spaced across the transmission band of the transceiver. One such arrangement is called discrete multi-tone (DMT) . DMT modems transmit data by dividing it into several interleaved bit streams, and using these bit streams to modulate several carriers.
Another application of multicarrier modulation is in OFDM systems, as described for instance in “OFDM for Wireless Multimedia Communications”, R. van Nee and R. Prasad, Artech House, 2000. Applications are for example, wireless LANs. This modulation technique also finds application in satellite communications, see for example, “Satellite Communications Systems”, G. Maral, M. Bousquet, Wiley, 1998.
In a receiver for use in an ADSL system for instance, following time domain equalization (TEQ), removal of the cycle prefix (CP), followed by a Fast Fourier Transform (FFT), complementary to the IFFT of the transmitter, is performed. The signal may then be passed to a frequency domain equalizer (FDEQ) to recover the signals as transmitted from the received signals, e.g., QAM symbols, from which the bit streams are recovered.
Calculating Fast Fourier transforms and their inverses can be performed by standard components and software packages. However, there is also a need to calculate fractional Fourier transforms in a more efficient manner.